CAREER: New directions in the study of zeros and moments of L-functions

职业：L 函数零点和矩研究的新方向

• 批准号: 2339274
• 负责人: Alexandra Florea
• 金额: $ 50万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2029-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2339274&HistoricalAwards=false
• 关键词: CAREER; New; directions; study; zeros

This project focuses on questions in analytic number theory, and concerns properties of the Riemann zeta-function and of more general L-functions. L-functions are functions on the complex plane that often encode interesting information about arithmetic objects, such as prime numbers, class numbers, or ranks of elliptic curves. For example, the Riemann zeta-function (which is one example of an L-function) is closely connected to the question of counting the number of primes less than a large number. Understanding the analytic properties of L-functions, such as the location of their zeros or their rate of growth, often provides insight into arithmetic questions of interest. The main goal of the project is to advance the knowledge of the properties of some families of L-functions and to obtain arithmetic applications. The educational component of the project involves groups of students at different stages, ranging from high school students to beginning researchers. Among the educational activities, the PI will organize a summer school in analytic number theory focusing on young mathematicians, and will run a yearly summer camp at UCI for talented high school students.At a more technical level, the project will investigate zeros of L-functions by studying their ratios and moments. While positive moments of L–functions are relatively well-understood, much less is known about negative moments and ratios, which have applications to many difficult questions in the field. The planned research will use insights from random matrix theory, geometry, sieve theory and analysis. The main goals fall under two themes. The first theme is developing a general framework to study negative moments of L-functions, formulating full conjectures and proving partial results about negative moments. The second theme involves proving new non-vanishing results about L-functions at special points. Values of L-functions at special points often carry important arithmetic information; the PI plans to show that wide classes of L-functions do not vanish at the central point (i.e., the center of the critical strip, where all the non-trivial zeros are conjectured to be), as well as to study correlations between the values of different L-functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目重点关注解析数论中的问题，并涉及黎曼 zeta 函数和更一般的 L 函数的属性，L 函数是复平面上的函数，通常编码有关算术对象的有趣信息，例如素数、例如，黎曼 zeta 函数（L 函数的一个示例）与计算小于大数的素数数量的问题密切相关。 L 函数的分析属性（例如零点位置或增长率）通常可以深入了解感兴趣的算术问题 该项目的主要目标是增进对某些 L 函数族属性的了解。该项目的教育部分涉及不同阶段的学生，从高中生到初级研究人员。在教育活动中，PI 将组织一个以年轻数学家为重点的解析数论暑期学校，并将运行在 UCI 为有才华的高中生举办的年度夏令营。在更技术的层面上，该项目将通过研究 L 函数的比率和矩来研究 L 函数的零点，虽然人们对 L 函数的正矩的了解相对较好，但了解得较少。已知的负矩和比率，可应用于该领域的许多难题。计划的研究将利用随机矩阵理论、几何、筛理论和分析的见解。研究负面时刻的总体框架L-函数，制定完整的猜想并证明关于负矩的部分结果。第二个主题涉及证明关于特殊点的 L-函数的新的非零结果。 PI 计划表明，广泛类别的 L 函数不会在中心点（即临界带的中心，所有非平凡零点都被推测在该处）处消失，以及研究不同 L 函数值之间的相关性。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。